From Newton’s method to Newton’s fractal (which Newton knew nothing about)

Who knew root-finding could be so complicated?
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Thanks to these viewers for their contributions to translations
German: Luatic
Hebrew: Omer Tuchfeld
Portuguese: luiz12apn

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Interactive for this video:

On fractal dimension:

Mathologer on the cubic formula:

Some articles on Newton's Fractal, and its cousins:

Some of the videos from this year's Summer of Math Exposition are fairly relevant to the topics covered here. Take a look at these ones,

The Beauty of Bézier Curves

The insolubility of the quintic:

The math behind rasterizing fonts:

Viewer-made interactive:

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These animations are largely made using a custom python library, manim. See the FAQ comments here:

You can find code for specific videos and projects here:

Music by Vincent Rubinetti.

Download the music on Bandcamp:

Stream the music on Spotify:

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Timestamps:
0:00 - Intro
0:48 - Roots of polynomials
5:55 - Newton’s method
11:16 - The fractal
17:56 - The boundary property
23:13 - Closing thoughts
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Various social media stuffs:

[diaz]

"What the %$!* is going on?"
—Pi creature, 2021.

After all of these years, the pi creature thingy finally expressed his anger against his master.

[loganhalstead]

"You can kinda eyeball what those values might be"
*Goes to 4 decimal places*

[DanaTheLateBloomingFruitLoop]

This approach to shorts is the best one I've come across. Great as always!

[itstrysten]

Thank you for saving me from doom scrolling

[thomasoltmann]

Showing a mandelbrot set emerge within the boundary of a Newton's fractal without explanation has got to be the biggest cliffhanger anyone ever put into a math video.

[dewaldstroebel]

Best quote of Grant ever:
"What the %$!* is going on here!?"

[maboesanman]

I escaped the shorts feed; thanks 3b1b.

[yandereyan]

Thank you for saving me from *T H E S H O R T S*

[fibby]

He has come back to us, armed with python and infinite math.

[CodeParade]

It's interesting to note that a fractal also appears when applying Newton's method to almost ANY function with multiple zeros, not just polynomials. Pretty much any system that is iterative and has some kind of instability (like a division that could be near zero in the Newton case) will form some kind of fractal.

[MisterAndyS]

It's crazy how fast computers are now, that this can be interactive! I wrote a program to display portions of the Mandelbrot set on my home computer in the early 1990s, probably 1024x768 resolution, and each render took several minutes.

[avanishverma]

I am pausing the video in middle to comment. I have tears in my eyes... just seeing the sheer beauty of it, I learnt Newton-Raphson method in my engineering without a slightest clue of what it meant. Now I am confident I can not only teach it but apply it too wherever necessary. Going back to the video now. Thank you for the great work you are doing.

[TheGrinningSkull]

The blobs on blobs example was so intuitive to help understand how the boundary could involve all roots throughout. You always explain things in such an amazing way!

[kristofsimoncic]

Three years ago the youtube algorithm did a great thing, and I for first time saw your video. Now I study applied mathematics on university. You, and your coworkers, have litteraly changed my life. I am thanking you, for showing me, that mathematics is more than some random numbers. I love your work and mathematics. I wish you good luck.🙏🙏🙏

[seb_]

Thanks for stopping my short binge, three blue one brown

[SpacemanCrag]

This video is stunningly beautiful in every way. I'm always amazed that each one of Grant's videos seems to be better than the last. It's genuinely inspiring.

[BadccVoid]

This is PHENOMENAL! Visualizing all of this makes it all the more fun.

[konstantinkh]

For practical cases with lots of roots, to avoid landing near these fractal boundaries, a good starting point is using Residue Theorem on 1/P(x) to find regions containing just one of the roots (or some smaller set of them, if there are degenerate roots) and then apply Newton-Raphson to finish it off. Generally, 1/P(x) is well-behaved exactly where Newton-Raphson isn't. Of course, if your objective function isn't a polynomial, that can go out the window too. The way you partition the region for computing contour integrals also matters, and there can certainly be difficult cases where it's hard not to drive a boundary through a pole. So a general algorithm can get quite complex, but it usually still relies on these two methods.

[royceaxle]

Thank you for helping me escape shorts.

[techmathmajor]

Absolutely love the video! This is a great presentation. One note: When discussing that Newton couldn't have known about these fractals for lack of a computer, there is an example of a classical (Western) mathematician who knew something about 'chaos' (and the complicated sets it creates) even in 1881. That's Henri Poincare when he discovered the homoclinic tangle while studying the 3 body problem. Such tangles are connected to Horseshoes (related to Smale's Horseshoe).